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  "body": "\n## Statement\n\nLet $p$ be a [[prime|01826KMG2H]].\nAn abelian $p$-group is an [[abelian group|082GHNPJ53]] in which every element has order a power of $p$.\nIt is *reduced* if its only [[divisible|ZXVBEPMTCN]] subgroup is $\\set{0}$.\nFor an abelian $p$-group $A$, define subgroups $p^\\alpha A$ for each [[ordinal|0YSCSX5YZC]] $\\alpha$ by $p^0 A = A$, $p^{\\alpha+1} A = p(p^\\alpha A)$, and $p^\\lambda A = \\bigcap_{\\alpha \u003c \\lambda} p^\\alpha A$ for limit ordinals $\\lambda$.\nLet $P_\\alpha = \\set{x \\in p^\\alpha A : px = 0}$, a vector space over the [[finite field|0ZXZ82FTX2]] $\\F_p$.\nThe *$\\alpha$-th Ulm invariant* of $A$ is $u_\\alpha(A) = \\dim_{\\F_p} (P_\\alpha / P_{\\alpha+1})$.\n\nLet $A$ and $B$ be [[countable|01TQB88H4C]] reduced abelian $p$-groups.\nThen $A \\cong B$ if and only if $u_\\alpha(A) = u_\\alpha(B)$ for every ordinal $\\alpha$.\n\n## Remarks\n\nIsomorphic groups have the same Ulm invariants, since an isomorphism carries each $P_\\alpha$ onto the corresponding subgroup.\nFor the converse one builds an isomorphism step by step, by a back-and-forth argument over the countably many elements.\nAt each step one keeps a partial isomorphism between finite subgroups that preserves the heights of elements.\nThe equality of the Ulm invariants is what makes each extension step possible.\n\nExample: for $A = \\Z/p^k\\Z$ with $k \\ge 1$, $P_n$ is cyclic of order $p$ for $n \u003c k$ and zero for $n \\ge k$, so $u_{k-1}(A) = 1$ and all other invariants vanish.\nFor a finite abelian $p$-group, $u_n$ counts the cyclic summands of order $p^{n+1}$, and the theorem reduces to the classification of finite abelian $p$-groups.\n\nCountability is necessary.\nLet $B = \\bigoplus_{n \\ge 1} \\Z/p^n\\Z$, and let $\\bar B$ be the torsion part of $\\prod_{n \\ge 1} \\Z/p^n\\Z$.\nBoth are reduced, both have $u_n = 1$ for every finite $n$ and $u_\\alpha = 0$ for $\\alpha \\ge \\omega$, but $B$ is countable and $\\bar B$ is uncountable, so they are not isomorphic.\n\n## Explanations\n\n### L1 human\n\nA countable abelian group in which every element has order a power of $p$ can be very complicated, with elements divisible by $p$ many times, infinitely many times, and beyond.\nUlm attached to such a group a list of numbers, indexed by ordinals, that measure how many independent elements of order $p$ sit at each level of divisibility.\nThe theorem says that, for countable groups with no divisible part, this list determines the group completely.\n\n### L2 human\n\nUlm's theorem extends the classification of finite abelian $p$-groups by cyclic decompositions to all countable reduced $p$-groups, where no cyclic decomposition need exist.\nA general countable abelian $p$-group is the direct sum of a [[divisible|ZXVBEPMTCN]] group and a reduced group.\nThe divisible part is a direct sum of [[Prüfer groups|F0XAMM2PPS]], determined by the number of summands.\nSo all countable abelian $p$-groups are classified.\nThe uncountable case has no such classification, as the example above shows.\n",
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